Problem: How many times do the graphs $r = 4 \cos \theta$ and $r = 8 \sin \theta$ intersect?
Explanation: For $r = 4 \cos \theta,$
\begin{align*}
x &= r \cos \theta = 4 \cos^2 \theta = 2 \cos 2 \theta + 2, \\
y &= r \sin \theta = 4 \sin \theta \cos \theta = 2 \sin 2 \theta.
\end{align*}Hence,
\[(x - 2)^2 + y^2 = 4 \cos^2 2 \theta + 4 \sin^2 2 \theta = 4.\]Thus, the graph of $r = 4 \cos \theta$ is the circle centered at $(2,0)$ with radius 2.

For $r = 8 \sin \theta,$
\begin{align*}
x &= r \cos \theta = 8 \sin \theta \cos \theta = 4 \sin 2 \theta, \\
y &= r \sin \theta = 8 \sin^2 \theta = 4 - 4 \cos 2 \theta.
\end{align*}Hence,
\[x^2 + (y - 4)^2 = 16 \sin^2 2 \theta + 16 \cos^2 2 \theta = 16.\]Thus, the graph of $r = 8 \sin \theta$ is the circle centered at $(0,4)$ with radius 4.

Plotting these circles, we find that they intersect at $\boxed{2}$ points.

[asy]
unitsize(0.5 cm);

pair moo (real t) {
  real r = 4*cos(t);
  return (r*cos(t), r*sin(t));
}

path foo = moo(0);
real t;

for (t = 0; t <= pi + 0.1; t = t + 0.1) {
  foo = foo--moo(t);
}

draw(foo,red);
label("$r = 4 \cos \theta$", (6.5,-1), red);

pair moo (real t) {
  real r = 8*sin(t);
  return (r*cos(t), r*sin(t));
}

path foo = moo(0);

for (t = 0; t <= pi + 0.1; t = t + 0.1) {
  foo = foo--moo(t);
}

draw(foo,blue);
label("$r = 8 \sin \theta$", (6.5,5), blue);

draw((-6,0)--(6,0));
draw((0,-2)--(0,10));

dot((2,0));
dot((0,4));
[/asy]